Misplaced Pages

#319680

38-490: Particle–Particle–Particle–Mesh ( PM ) is a Fourier-based Ewald summation method to calculate potentials in N-body simulations . The potential could be the electrostatic potential among N point charges i.e. molecular dynamics , the gravitational potential among N gas particles in e.g. smoothed particle hydrodynamics , or any other useful function. It is based on the particle mesh method, where particles are interpolated onto

76-549: A 1 − n 2 a 2 − n 3 a 3 ) {\displaystyle L(\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n_{1},n_{2},n_{3}}\delta (\mathbf {r} -n_{1}\mathbf {a} _{1}-n_{2}\mathbf {a} _{2}-n_{3}\mathbf {a} _{3})} Since this is a convolution , the Fourier transformation of ρ TOT ( r ) {\displaystyle \rho _{\text{TOT}}(\mathbf {r} )}

114-480: A 1 − n 2 a 2 − n 3 a 3 ) {\displaystyle \rho _{\text{TOT}}(\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n_{1},n_{2},n_{3}}\sum _{\mathrm {charges} \ k}q_{k}\delta (\mathbf {r} -\mathbf {r} _{k}-n_{1}\mathbf {a} _{1}-n_{2}\mathbf {a} _{2}-n_{3}\mathbf {a} _{3})} Here, δ ( x ) {\displaystyle \delta (\mathbf {x} )}

152-1222: A 1 ⋅ ( a 2 × a 3 ) {\displaystyle \Omega \ {\stackrel {\mathrm {def} }{=}}\ \mathbf {a} _{1}\cdot \left(\mathbf {a} _{2}\times \mathbf {a} _{3}\right)} is the volume of the central unit cell (if it is geometrically a parallelepiped , which is often but not necessarily the case). Note that both L ( r ) {\displaystyle L(\mathbf {r} )} and L ~ ( k ) {\displaystyle {\tilde {L}}(\mathbf {k} )} are real, even functions. For brevity, define an effective single-particle potential v ( r )   = d e f   ∫ d r ′ ρ u c ( r ′ )   φ ℓ r ( r − r ′ ) {\displaystyle v(\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} ^{\prime }\,\rho _{uc}(\mathbf {r} ^{\prime })\ \varphi _{\ell r}(\mathbf {r} -\mathbf {r} ^{\prime })} Since this

190-438: A r g e s   k q k δ ( r − r k ) {\displaystyle \rho _{uc}(\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{\mathrm {charges} \ k}q_{k}\delta (\mathbf {r} -\mathbf {r} _{k})} and the total charge density field ρ TOT ( r ) {\displaystyle \rho _{\text{TOT}}(\mathbf {r} )}

228-481: A Gaussian distribution . The method assumes that the short-range part can be summed easily; hence, the problem becomes the summation of the long-range term. Due to the use of the Fourier sum, the method implicitly assumes that the system under study is infinitely periodic (a sensible assumption for the interiors of crystals). One repeating unit of this hypothetical periodic system is called a unit cell . One such cell

266-471: A convolution of ρ u c ( r ) {\displaystyle \rho _{uc}(\mathbf {r} )} with a lattice function L ( r ) {\displaystyle L(\mathbf {r} )} L ( r )   = d e f   ∑ n 1 , n 2 , n 3 δ ( r − n 1

304-401: A direct sum E s r {\displaystyle E_{sr}} of the short-ranged potential in real space E s r = ∑ i , j φ s r ( r j − r i ) {\displaystyle E_{sr}=\sum _{i,j}\varphi _{sr}(\mathbf {r} _{j}-\mathbf {r} _{i})} (this is

342-451: A grid, and the potential is solved for this grid (e.g. by solving the discrete Poisson equation ). This interpolation introduces errors in the force calculation, particularly for particles that are close together. Essentially, the particles are forced to have a lower spatial resolution during the force calculation. The PM algorithm attempts to remedy this by calculating the potential through a direct sum for particles that are close, and through

380-529: A shell of radius R {\displaystyle R} grows like R 2 {\textstyle R^{2}} ; (2) the strength of a single dipole-dipole interaction falls like 1 / R 3 {\textstyle 1/{R^{3}}} ; and (3) the mathematical summation ∑ n = 1 ∞ 1 n {\textstyle \sum _{n=1}^{\infty }{\frac {1}{n}}} diverges. This somewhat surprising result can be reconciled with

418-524: Is a poorly defined unit cell, which must be charge neutral to avoid infinite sums. Ewald summation was developed as a method in theoretical physics , long before the advent of computers . However, the Ewald method has enjoyed widespread use since the 1970s in computer simulations of particle systems, especially those whose particles interact via an inverse square force law such as gravity or electrostatics . Recently, PME has also been used to calculate

SECTION 10

#1732776431320

456-409: Is a product ρ ~ TOT ( k ) = L ~ ( k ) ρ ~ u c ( k ) {\displaystyle {\tilde {\rho }}_{\text{TOT}}(\mathbf {k} )={\tilde {L}}(\mathbf {k} ){\tilde {\rho }}_{uc}(\mathbf {k} )} where the Fourier transform of the lattice function

494-499: Is also a convolution, the Fourier transformation of the same equation is a product V ~ ( k )   = d e f   ρ ~ u c ( k ) Φ ~ ( k ) {\displaystyle {\tilde {V}}(\mathbf {k} )\ {\stackrel {\mathrm {def} }{=}}\ {\tilde {\rho }}_{uc}(\mathbf {k} ){\tilde {\Phi }}(\mathbf {k} )} where

532-660: Is another sum over delta functions L ~ ( k ) = ( 2 π ) 3 Ω ∑ m 1 , m 2 , m 3 δ ( k − m 1 b 1 − m 2 b 2 − m 3 b 3 ) {\displaystyle {\tilde {L}}(\mathbf {k} )={\frac {\left(2\pi \right)^{3}}{\Omega }}\sum _{m_{1},m_{2},m_{3}}\delta (\mathbf {k} -m_{1}\mathbf {b} _{1}-m_{2}\mathbf {b} _{2}-m_{3}\mathbf {b} _{3})} where

570-945: Is chosen as the "central cell" for reference and the remaining cells are called images . The long-range interaction energy is the sum of interaction energies between the charges of a central unit cell and all the charges of the lattice. Hence, it can be represented as a double integral over two charge density fields representing the fields of the unit cell and the crystal lattice E ℓ r = ∬ d r d r ′ ρ TOT ( r ) ρ u c ( r ′ )   φ ℓ r ( r − r ′ ) {\displaystyle E_{\ell r}=\iint d\mathbf {r} \,d\mathbf {r} ^{\prime }\,\rho _{\text{TOT}}(\mathbf {r} )\rho _{uc}(\mathbf {r} ^{\prime })\ \varphi _{\ell r}(\mathbf {r} -\mathbf {r} ^{\prime })} where

608-429: Is normally accomplished by deliberately constructing a charge-neutral unit cell that can be infinitely "tiled" to form images; however, to properly account for the effects of this approximation, these images are reincorporated back into the original simulation cell. The overall effect is called a periodic boundary condition . To visualize this most clearly, think of a unit cube; the upper face is effectively in contact with

646-725: Is the Dirac delta function , a 1 {\displaystyle \mathbf {a} _{1}} , a 2 {\displaystyle \mathbf {a} _{2}} and a 3 {\displaystyle \mathbf {a} _{3}} are the lattice vectors and n 1 {\displaystyle n_{1}} , n 2 {\displaystyle n_{2}} and n 3 {\displaystyle n_{3}} range over all integers. The total field ρ TOT ( r ) {\displaystyle \rho _{\text{TOT}}(\mathbf {r} )} can be represented as

684-516: Is the same sum over the unit-cell charges q k {\displaystyle q_{k}} and their periodic images ρ TOT ( r )   = d e f   ∑ n 1 , n 2 , n 3 ∑ c h a r g e s   k q k δ ( r − r k − n 1

722-887: Is the surface normal vector and P {\displaystyle \mathbf {P} } represents the net dipole moment per volume. The interaction energy U {\displaystyle U} of the dipole in a central unit cell with that surface charge density can be written U = 1 2 V u c ∫ ( p u c ⋅ r ) ( p u c ⋅ n ) r 3 d S {\displaystyle U={\frac {1}{2V_{uc}}}\int {\frac {\left(\mathbf {p} _{uc}\cdot \mathbf {r} \right)\left(\mathbf {p} _{uc}\cdot \mathbf {n} \right)}{r^{3}}}\,dS} where p u c {\displaystyle \mathbf {p} _{uc}} and V u c {\displaystyle V_{uc}} are

760-434: Is to replace the direct summation of interaction energies between point particles E TOT = ∑ i , j φ ( r j − r i ) = E s r + E ℓ r {\displaystyle E_{\text{TOT}}=\sum _{i,j}\varphi (\mathbf {r} _{j}-\mathbf {r} _{i})=E_{sr}+E_{\ell r}} with two summations,

798-851: The r − 6 {\displaystyle r^{-6}} part of the Lennard-Jones potential in order to eliminate artifacts due to truncation. Applications include simulations of plasmas , galaxies and molecules . In the particle mesh method, just as in standard Ewald summation, the generic interaction potential is separated into two terms φ ( r )   = d e f   φ s r ( r ) + φ ℓ r ( r ) {\displaystyle \varphi (\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \varphi _{sr}(\mathbf {r} )+\varphi _{\ell r}(\mathbf {r} )} . The basic idea of particle mesh Ewald summation

SECTION 20

#1732776431320

836-504: The Fourier transforms of the potential and the charge density (this is the Ewald part). Since both summations converge quickly in their respective spaces (real and Fourier), they may be truncated with little loss of accuracy and great improvement in required computational time. To evaluate the Fourier transform ρ ~ ( k ) {\displaystyle {\tilde {\rho }}(\mathbf {k} )} of

874-773: The particle part of particle mesh Ewald ) and a summation in Fourier space of the long-ranged part E ℓ r = ∑ k Φ ~ ℓ r ( k ) | ρ ~ ( k ) | 2 {\displaystyle E_{\ell r}=\sum _{\mathbf {k} }{\tilde {\Phi }}_{\ell r}(\mathbf {k} )\left|{\tilde {\rho }}(\mathbf {k} )\right|^{2}} where Φ ~ ℓ r {\displaystyle {\tilde {\Phi }}_{\ell r}} and ρ ~ ( k ) {\displaystyle {\tilde {\rho }}(\mathbf {k} )} represent

912-722: The Fourier transform is defined V ~ ( k ) = ∫ d r   v ( r )   e − i k ⋅ r {\displaystyle {\tilde {V}}(\mathbf {k} )=\int d\mathbf {r} \ v(\mathbf {r} )\ e^{-i\mathbf {k} \cdot \mathbf {r} }} The energy can now be written as a single field integral E ℓ r = ∫ d r   ρ TOT ( r )   v ( r ) {\displaystyle E_{\ell r}=\int d\mathbf {r} \ \rho _{\text{TOT}}(\mathbf {r} )\ v(\mathbf {r} )} Using Plancherel theorem ,

950-532: The charge density field efficiently, one uses the fast Fourier transform , which requires that the density field be evaluated on a discrete lattice in space (this is the mesh part). Due to the periodicity assumption implicit in Ewald summation, applications of the PME method to physical systems require the imposition of periodic symmetry. Thus, the method is best suited to systems that can be simulated as infinite in spatial extent. In molecular dynamics simulations this

988-603: The definition of r {\displaystyle \mathbf {r} } , which points towards the charge, not away from it. The Ewald summation was developed by Paul Peter Ewald in 1921 (see References below) to determine the electrostatic energy (and, hence, the Madelung constant ) of ionic crystals. Generally, different Ewald summation methods give different time complexities . Direct calculation gives O ( N 2 ) {\displaystyle O(N^{2})} , where N {\displaystyle N}

1026-468: The density field to a mesh makes the PME method more efficient for systems with "smooth" variations in density, or continuous potential functions. Localized systems or those with large fluctuations in density may be treated more efficiently with the fast multipole method of Greengard and Rokhlin. The electrostatic energy of a polar crystal (i.e. a crystal with a net dipole p u c {\displaystyle \mathbf {p} _{uc}} in

1064-1666: The energy can also be summed in Fourier space E ℓ r = ∫ d k ( 2 π ) 3   ρ ~ TOT ∗ ( k ) V ~ ( k ) = ∫ d k ( 2 π ) 3 L ~ ∗ ( k ) | ρ ~ u c ( k ) | 2 Φ ~ ( k ) = 1 Ω ∑ m 1 , m 2 , m 3 | ρ ~ u c ( k ) | 2 Φ ~ ( k ) {\displaystyle E_{\ell r}=\int {\frac {d\mathbf {k} }{\left(2\pi \right)^{3}}}\ {\tilde {\rho }}_{\text{TOT}}^{*}(\mathbf {k} ){\tilde {V}}(\mathbf {k} )=\int {\frac {d\mathbf {k} }{\left(2\pi \right)^{3}}}{\tilde {L}}^{*}(\mathbf {k} )\left|{\tilde {\rho }}_{uc}(\mathbf {k} )\right|^{2}{\tilde {\Phi }}(\mathbf {k} )={\frac {1}{\Omega }}\sum _{m_{1},m_{2},m_{3}}\left|{\tilde {\rho }}_{uc}(\mathbf {k} )\right|^{2}{\tilde {\Phi }}(\mathbf {k} )} where k = m 1 b 1 + m 2 b 2 + m 3 b 3 {\displaystyle \mathbf {k} =m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}} in

1102-410: The final summation. This is the essential result. Once ρ ~ u c ( k ) {\displaystyle {\tilde {\rho }}_{uc}(\mathbf {k} )} is calculated, the summation/integration over k {\displaystyle \mathbf {k} } is straightforward and should converge quickly. The most common reason for lack of convergence

1140-399: The finite energy of real crystals because such crystals are not infinite, i.e. have a particular boundary. More specifically, the boundary of a polar crystal has an effective surface charge density on its surface σ = P ⋅ n {\displaystyle \sigma =\mathbf {P} \cdot \mathbf {n} } where n {\displaystyle \mathbf {n} }

1178-928: The infinitesimal electric field generated by an infinitesimal surface charge d q   = d e f   σ d S {\displaystyle dq\ {\stackrel {\mathrm {def} }{=}}\ \sigma dS} ( Coulomb's law ) d E   = d e f   ( − 1 4 π ϵ ) d q   r r 3 = ( − 1 4 π ϵ ) σ d S   r r 3 {\displaystyle d\mathbf {E} \ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {-1}{4\pi \epsilon }}\right){\frac {dq\ \mathbf {r} }{r^{3}}}=\left({\frac {-1}{4\pi \epsilon }}\right){\frac {\sigma \,dS\ \mathbf {r} }{r^{3}}}} The negative sign derives from

P3M - Misplaced Pages Continue

1216-403: The lower face, the right with the left face, and the front with the back face. As a result, the unit cell size must be carefully chosen to be large enough to avoid improper motion correlations between two faces "in contact", but still small enough to be computationally feasible. The definition of the cutoff between short- and long-range interactions can also introduce artifacts. The restriction of

1254-578: The net dipole moment and volume of the unit cell, d S {\displaystyle dS} is an infinitesimal area on the crystal surface and r {\displaystyle \mathbf {r} } is the vector from the central unit cell to the infinitesimal area. This formula results from integrating the energy d U = − p u c ⋅ d E {\displaystyle dU=-\mathbf {p} _{uc}\cdot d\mathbf {E} } where d E {\displaystyle d\mathbf {E} } represents

1292-791: The particle mesh method for particles that are separated by some distance. This computational physics -related article is a stub . You can help Misplaced Pages by expanding it . Ewald summation Ewald summation rewrites the interaction potential as the sum of two terms, φ ( r )   = d e f   φ s r ( r ) + φ ℓ r ( r ) , {\displaystyle \varphi (\mathbf {r} )\ {\stackrel {\mathrm {def} }{=}}\ \varphi _{sr}(\mathbf {r} )+\varphi _{\ell r}(\mathbf {r} ),} where φ s r ( r ) {\displaystyle \varphi _{sr}(\mathbf {r} )} represents

1330-497: The reciprocal space vectors are defined b 1   = d e f   2 π a 2 × a 3 Ω {\displaystyle \mathbf {b} _{1}\ {\stackrel {\mathrm {def} }{=}}\ 2\pi {\frac {\mathbf {a} _{2}\times \mathbf {a} _{3}}{\Omega }}} (and cyclic permutations) where Ω   = d e f  

1368-437: The short-range term whose sum quickly converges in real space and φ ℓ r ( r ) {\displaystyle \varphi _{\ell r}(\mathbf {r} )} represents the long-range term whose sum quickly converges in Fourier (reciprocal) space. The long-ranged part should be finite for all arguments (most notably r  = 0) but may have any convenient mathematical form, most typically

1406-419: The unit cell) is conditionally convergent , i.e. depends on the order of the summation. For example, if the dipole-dipole interactions of a central unit cell with unit cells located on an ever-increasing cube, the energy converges to a different value than if the interaction energies had been summed spherically. Roughly speaking, this conditional convergence arises because (1) the number of interacting dipoles on

1444-507: The unit-cell charge density field ρ u c ( r ) {\displaystyle \rho _{uc}(\mathbf {r} )} is a sum over the positions r k {\displaystyle \mathbf {r} _{k}} of the charges q k {\displaystyle q_{k}} in the central unit cell ρ u c ( r )   = d e f   ∑ c h

#319680